Weakly-Initial Set of Covers

Definition

A cover of a set $Y$ is just a surjective map from another set $X$:

$$\mathrm{Cov}(Y)=\sum _{X:U}X\twoheadrightarrow Y$$ Link to original

Definition

The category of covers of $Y$ is given as:

• Objects are covers in $\mathrm{Cov}(Y)$
• $\mathrm{Hom}[(X,p),(Z,q)]={\sum }_{f:X\to Z}q\circ f=p$

Link to original

Definition

A weakly-initial set in an arbitrary category with families $\mathcal{C}$ is a family $(I,\bar{X})$ such that:

$$\forall Z:\mathcal{C}.\exists i\in I.\exists f:X_i\to Z$$Link to original

Definition

A weakly initial set of covers over $Y$ is just a weakly initial set in the category of covers $\mathbf{Cov}(Y)$.

Suppose $(I,\bar{X},\bar{p})$ is a family of covers $\bar{X}:I\to \mathcal{U};\bar{p}:{\prod }_{i:I}{X}_i\twoheadrightarrow Y$ for some $I:\mathcal{U}$. Say $(I,\bar{X},\bar{p})$ is WISC for $Y$ iff every cover $q$ can be factored through some $p_i$:

$$\begin{array}{rl}& \forall q:Z\twoheadrightarrow Y\\ & \exists i:I;f:X_i\to Z\\ & \text{s.t. }{p}_i=q\circ f\end{array}$$

Remarks

Full choice, such as ZFC implies that identities are a weakly-initial object of $\mathrm{Cov}(Y)$. Choice tells that all surjections have a section.

AC $\Rightarrow$ WISC

In the presence of full choice, every set covers itself. $(Y,id)$ is a weakly initial object in the category of $Z$ covers. Given any $Y$ cover $(Z,q)$, then since surjections splitting is exactly what choice gives us: a witness from every fibre.

Independence from ZF

Since AC implies WISC, the consistency of ZFC implies ZF is consistent with WISC. It has been proven that WISC is not entailed by ZF.

The proof works by contradiction in a model where all limit ordinals have countable cofinality:

1. Given a WISC over $\mathbb{N}$, we obtain a set-sized container $B:A\to \mathcal{U}$ representing these covers.
2. Extend it by adding $0$ and $1$ for zero and successor constructors to get $B^{\prime }:A^{\prime }\to \mathcal{U}$.
3. Let $W$ be the W type of $A^{\prime }◃B^{\prime }$.
4. Define $m:W\to \mathrm{Ord}$ by transfinite induction s.t.

m(\sup~0~\_) &= 0\\ m(\sup~1~t) &= 1+t(x)\\ m(\sup~a~t) &= \sup \{m(t(b)) \mid b:B(a)\}\qquad\text{ if } a\in A \end{aligned}$$ 5. This W-type generates a set of ordinals closed under limits of cofinality $\omega$. 6. We appeal to the result by [[gitik1980-cofinality-omega]] which establishes the consistency of the statement *all limit ordinals have cofinality $\omega$* with ZF. 7. If all limit ordinals have cofinality $\omega$, every ordinal is reached by $m$. Because $W$ is a set, the proper class of all ordinals would be bounded by a set, contradicting the Burali-Forti paradox. ## References * [[gitik1980-cofinality-omega]] * [[berg2012-wisc]]